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Fronzbot
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This question has been giving my brain some trouble. I think i have the Electric Field integral calculated correctly (but I'll post that up here too just in case I'm wrong which is likely) but when I get down to finding the charge on the rod I literally am making a wild guess at the solution.
A rod with length L has a charge density of Ax where A is a constant and x is measured from the center of the rod, with positive to the right. Set up the integral to find the electric field at point P (which is at a distance h above the center of the rod) but do not evaluate. Be sure to tell the direction of the net electric field. Also find the total charge on the rod.
dEp = kdq/r2 where k = 8.99E9
So for the integral, I set dq = Ax*dx and r = [tex]\sqrt{L^2/4 + h^2}[/tex] I stated that, due to symmetry, the net electric field on the x-axis should be equal to 0, though I don't know if I can make that assumption since I don't know the charge on the rod? Regardless, I ended up getting this integral:
Ep = kA/(L[tex]^2[/tex]/4 + h[tex]^2[/tex]) [tex]\int { xdx}[/tex] in the +y direction. The integral is from 0 to h, as well. Not sure if those are the correct bounds to be using, but it made sense to me.
From there I figured the only way I could calculate the charge is by solving the integral and then solving for Q:
Ep = kAx[tex]^2[/tex]/(L[tex]^2[/tex]/2 + h[tex]^2[/tex]/2) from 0 to h
Ep = 2kAh[tex]^2[/tex]/(L[tex]^2[/tex] + h[tex]^2[/tex]) and since [tex]\lambda[/tex] = Ax and the units for [tex]\lambda[/tex] = N/C you can figure out that the units for A must be C/m[tex]^2[/tex] so A = Q/L[tex]^2[/tex]
Ep = 2kQh[tex]^2[/tex]/(L[tex]^4[/tex] + h[tex]^2[/tex]*L[tex]^2[/tex])
L[tex]^4[/tex] + h[tex]^2[/tex]*L[tex]^2[/tex] = 2kQh[tex]^2[/tex]
Q = L[tex]^4[/tex] + h^2[tex]^2[/tex]*L[tex]^2[/tex]/2kh[tex]^2[/tex]
=================================
So I am almost 100% sure that the way I solved for charge will make any physicist cry because, honestly, it just looks wrong. Besides, I believe that the way I solved this here also assumes that the Electric Field at P is equal to 1. If anyone can help me through this problem and help me understand my mistakes I would greatly appreciate it. It's weird that this problem is giving me trouble when other things like charged arcs are easy for me to grasp.
Thanks in advance!
-Kevin
Homework Statement
A rod with length L has a charge density of Ax where A is a constant and x is measured from the center of the rod, with positive to the right. Set up the integral to find the electric field at point P (which is at a distance h above the center of the rod) but do not evaluate. Be sure to tell the direction of the net electric field. Also find the total charge on the rod.
Homework Equations
dEp = kdq/r2 where k = 8.99E9
The Attempt at a Solution
So for the integral, I set dq = Ax*dx and r = [tex]\sqrt{L^2/4 + h^2}[/tex] I stated that, due to symmetry, the net electric field on the x-axis should be equal to 0, though I don't know if I can make that assumption since I don't know the charge on the rod? Regardless, I ended up getting this integral:
Ep = kA/(L[tex]^2[/tex]/4 + h[tex]^2[/tex]) [tex]\int { xdx}[/tex] in the +y direction. The integral is from 0 to h, as well. Not sure if those are the correct bounds to be using, but it made sense to me.
From there I figured the only way I could calculate the charge is by solving the integral and then solving for Q:
Ep = kAx[tex]^2[/tex]/(L[tex]^2[/tex]/2 + h[tex]^2[/tex]/2) from 0 to h
Ep = 2kAh[tex]^2[/tex]/(L[tex]^2[/tex] + h[tex]^2[/tex]) and since [tex]\lambda[/tex] = Ax and the units for [tex]\lambda[/tex] = N/C you can figure out that the units for A must be C/m[tex]^2[/tex] so A = Q/L[tex]^2[/tex]
Ep = 2kQh[tex]^2[/tex]/(L[tex]^4[/tex] + h[tex]^2[/tex]*L[tex]^2[/tex])
L[tex]^4[/tex] + h[tex]^2[/tex]*L[tex]^2[/tex] = 2kQh[tex]^2[/tex]
Q = L[tex]^4[/tex] + h^2[tex]^2[/tex]*L[tex]^2[/tex]/2kh[tex]^2[/tex]
=================================
So I am almost 100% sure that the way I solved for charge will make any physicist cry because, honestly, it just looks wrong. Besides, I believe that the way I solved this here also assumes that the Electric Field at P is equal to 1. If anyone can help me through this problem and help me understand my mistakes I would greatly appreciate it. It's weird that this problem is giving me trouble when other things like charged arcs are easy for me to grasp.
Thanks in advance!
-Kevin